[Proof of Theorem REF ]
Note that, by Lemma 1 in [1]}, the total variation of \(\theta \)  wrt to the \(L_0\) -norm in the DFS-chain graph \(G_C\)  is bounded by at most twice that in \(G\) , that is, \(\theta \in l_0[G,s]\)  always implies that \(\theta \in l_0[2s].\)  Also, assumptions REF  - REF  for an arbitrary graph \(G\)  implies assumptions REF  - REF  for a linear chain graph. Hence the result readily follows from that of Theorem REF .
